Inverting Stiffness and Damping for Hydraulic Conductivity

Octave program KD4kvmb.m can be used to invert values of stiffness and damping to hydraulic conductivity. For example, in a down-hole shear wave transmission survey, one can measure body wave dispersion and amplitude decay with distance to obtain values for stiffness and damping. The procedure to invert velocity dispersion and amplitude decay is detailed in Michaels (10). The result of the inversion are two coefficients of the viscoelastic 1D wave equation (beam divergence is corrected for in the process). The wave equation is a 3rd order PDE:

$\displaystyle \dfrac{\partial ^2 u}{\partial t^2}=C_1 \dfrac{\partial ^2 u}{\partial z ^2}
+ C_2 \dfrac{\partial ^3 u}{\partial t \partial z^2}$ (53)

where $C_1 $ is stiffness ($m^2/s^2$) and $C_2 $ is damping ($m^2/s$). The ratio $C_2 / C_1 $ yields relaxation time in seconds. In equation 53 time is represented by $t$ and particle displacement by $u$. Distance in the direction of wave propagation is $z$. Figure 69 shows an example run assuming shaking at 12 Hz and a porosity of 25%. Note that if the value of $C_2 $ is so large compared to $C_1 $, that it produces a KV damping ratio that falls above the peak of the curve, no solution is possible. In such a case, one may re-evaluate all the assumptions. When deciding on a likely solution from the two possible, most soils will fall on the coupled side of the curve, making this example solution of $K_d = .0122 m/s $ the more likely one.


Figure 69: Octave program, KD4kvmb.m prompts the user for porosity (n), stiffness ($C_1 $), damping ($C_2 $), frequency of shaking and related uncertainties. Then when run, a display of the solution is given in a message box. Also show is the graphical image of the process. The $C_1 $ and $C_2 $ values produce a KV damping ratio that is represented by the horizontal line that intersects the KVMB to KV curve. The two intersections are the solution.
\includegraphics[scale=0.9]{FigureTT}

Are these predictions accurate? No, of course not. But they are a starting point in estimating hydraulic conductivity from shear wave measurements. While the predicted behaviors are likely correct in terms of how real soils behave, real soils are far more complex, pore spaces are not cylindrical tubes, and fluid flow is not always laminar. In the context of granular soils, these tools may be helpful in mapping soil units, their permeabilities, and predicting levels of damping that might occur when exposed to seismic waves.